Jean Richards-Houck makes her way toward Republican presidential candidate, former House Speaker Newt Gingrich's bus during his visit to a polling place at Celebration Heritage Hall in Celebration, Fla., Tuesday, Jan. 31, 2012. (AP Photo/Matt Rourke)

TAMPA, Fla. — Mitt Romney routed Newt Gingrich in the Florida primary Tuesday night, rebounding smartly from an earlier defeat and taking a major step toward the Republican presidential nomination. Despite the one-sided setback, the former House speaker vowed to press on.

Romney, talking unity like a nominee, said he was ready “to lead this party and our nation.” In remarks to cheering supporters, the former Massachusetts governor unleashed a strong attack on President Barack Obama and said the competitive fight for the GOP nomination “does not divide us, it prepares us” for the fall.

“Mr. President, you were elected to lead, you chose to follow, and now it’s time to get out of the way, he declared.

Returns from 79 percent of Florida’s precincts showed Romney with 47 percent of the vote, to 32 percent for Gingrich.

For the first time in the campaign, exit polls showed a gender gap, and it worked to Romney’s advantage.

He was leading Gingrich 51-29 percent among women voters, and was winning men by a far smaller margin of 41-36.

Ominously for the thrice-married Gingrich, only about half of women voters said they had a favorable view of him as a person, compared to about eight in 10 for Romney.

As in Iowa, New Hampshire and South Carolina, about half of Florida primary voters said the most important factor for them was backing a candidate who could defeat Obama in November, according to exit poll results conducted for The Associated Press and the television networks.

Not surprisingly, in a state with an unemployment rate hovering around 10 percent, about two-thirds of voters said the economy was their top issue. More than eight in 10 said they were falling behind or just keeping up. And half said that home foreclosures have been a major problem in their communities.

The winner-take-all primary was worth 50 Republican National Convention delegates, by far the most of any primary state so far. That gave Romney a total of 87, to 26 for Gingrich, 14 for Santorum and four for Paul, with 1,144 required to clinch the nomination.

But the bigger prize was precious political momentum in the race to pick an opponent for Obama in a nation struggling to recover from the deepest recession in decades.

That belonged to Romney when he captured the New Hampshire primary three weeks ago, then swung stunningly to Gingrich when he countered with a South Carolina upset 11 days later.

Now it was back with the former Massachusetts governor, after a 10-day comeback marked by a change to more aggressive tactics, coupled with an efficient use of an overwhelming financial advantage to batter Gingrich in television commercials.

Gingrich brushed aside any talk of quitting the race.

“We are going to contest everyplace,” he said, standing in front of a sign that read “46 states to go.”

“It is now clear that this will be a two person race between the conservative leader, Newt Gingrich, and the Massachusetts moderate,” he said.

The race now turns to Nevada, where Romney won the state’s caucuses four years ago and is favored to repeat his triumph this Saturday. Caucuses in Colorado, Minnesota and Maine follow, with primaries in Wisconsin on Feb. 21 and in Michigan and Arizona at the end of the month.

Gingrich, from neighboring Georgia, swept into Florida from South Carolina, only to run headlong into a different Romney from the one he had left in his wake in South Carolina.

Romney, the former Massachusetts governor, shed his reluctance to attack Gingrich, the former House speaker, unleashing hard-hitting ads on television, sharpening his performance in a pair of debates and deploying surrogates to the edges of Gingrich’s own campaign appearances, all in hopes of unnerving him.

Restore our Future, an outside group supporting Romney, accounted for about $8.8 million in the ad wars, and the candidate and the “super PAC” combined outspent Gingrich and Winning The Future, the organization backing him, by about $15.5 million to $3.3 million, an advantage of nearly 5-1.

Gingrich responded by assailing Romney as a man incapable of telling the truth and vowed to remain in the race until the Republican National Convention next summer. He won the endorsement of campaign dropout Herman Cain and increasingly sought the support of evangelicals and tea party advocates, a former House speaker running as the anti-establishment insurgent of the party he once helped lead.

Bombarded by harsh television advertising, some Floridians said they had soured on both candidates.

“The dirty ads really turned me off on Mitt Romney,” said Dorothy Anderson, of Pinellas Park, adding she was voting for Gingrich. She said of Romney, “In fact if he gets the nomination, I probably won’t vote for him.”

At the same polling place, Romney supporter Curtis Dempsey expressed similar feelings but about Gingrich. “The only thing Newt Gingrich has to offer is a big mouth,” he said.

Voters frequently say they are offended or appalled by negative ads. But polls show consistently that the commercials are able to sway the opinions of large numbers of voters, and they are a staple of nearly all campaigns.

Santorum had no money for television ads to back up his strong debate performances. He left the state at one point, saying he was going home to Pennsylvania to prepare his income tax returns. But he stayed longer than anticipated, because of the hospitalization of his 3-year-old daughter with pneumonia. The girl has a rare genetic disorder, Trisomy 18.

Santorum was already in Nevada Tuesday night, campaigning for the state’s caucuses on Saturday. “Newt Gingrich had his chance. He had his shot,” he said. Now Republicans are “looking for a different conservative.”

Santorum and Paul both also campaigned in Colorado on Tuesday as Florida Republicans were voting. The state has caucuses on Feb 7, the same day as Minnesota.

Even before that come caucuses in Nevada, a state that Romney won when he sought the nomination in 2008 and is favored to capture again.

By contrast, both Romney and Gingrich campaigned across Florida on primary day as the polls opened.

Exuding confidence, the former Massachusetts governor said — even though the figures said otherwise — that he had been outspent in South Carolina.

“I needed to make sure that instead of being outgunned in terms of attacks, that I responded aggressively, and hopefully that will have served me well here,” he told reporters.

Gingrich, combative as usual, said the race for the nomination won’t be decided until summer, “unless Romney drops out.”

More than 600,000 Floridians voted before the polls opened, either by absentee or early ballot, exceeding the figure from four years ago, and raising the possibility of a record turnout. Vote totals in the three contests to date — Iowa, New Hampshire and South Carolina — have all been records.